Tests and CIs The P-value for a one-sided test of the null...

Tests and CIs The P-value for a one-sided test of the null hypothesis $H_{0} : \mu=15$ is 0.03 .
(a) Does the 99$\%$ confidence interval for $\mu$ include 15$?$ Why or why not?
(b) Does the 95$\%$ confidence interval for $\mu$ include 15? Why or why not?

Key Concepts

Relationship Between Hypothesis Testing and Confidence Intervals

Hypothesis testing and confidence intervals are closely related; if a hypothesized value is not contained in the (1 ? ?)100% confidence interval, it would typically be rejected by a two-sided hypothesis test at the ? significance level. However, this relationship can differ for one-sided tests where the rejection region is not symmetric.

Confidence Level

The confidence level (such as 95% or 99%) represents the proportion of confidence intervals that would contain the true parameter if the same sampling procedure were repeated many times. Higher confidence levels produce wider intervals because they need to cover the true parameter more reliably.

One-Sided vs Two-Sided Tests

One-sided tests assess hypotheses in only one direction (either greater than or less than a certain value), which leads to different critical values and interpretations compared to two-sided tests. Confidence intervals are generally constructed for two-sided tests, and comparing them with results from one-sided tests requires careful consideration of the differences in their designs.

P-value

A P-value is the probability of obtaining a test statistic at least as extreme as what was observed, assuming that the null hypothesis is true. It is used to quantify the strength of evidence against the null hypothesis in hypothesis testing.

Confidence Interval

A confidence interval is a range of values derived from sample data that is likely to contain the true value of the parameter with a specified level of confidence. It provides an estimated range of plausible values for the population parameter.

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